**2. Formulation**

We consider a third grade stagnation point flow towards a radiative stretching cylinder in the context of double stratification. The stagnation point is discussed categorically. Analysis is scrutinized in the presence of Thermophoresis, Brownian diffusion, double stratification and heat source/sink. Suitable typical transformations are used to drive the system of ordinary differential equation. The governing system is subjected to optimal homotopy analysis method (OHAM) for convergen<sup>t</sup> series solutions.. The effect of double stratification and thermal radiation is accounted. We assume that z-axis is directed along the given stretching cylinder whereas the radial r-axis goes perpendicular to it. Thus,

$$\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0,\tag{1}$$

*ρ u ∂w ∂r* + *w ∂w ∂z* = *We dwe dz* + *μ ∂*2*w ∂r*<sup>2</sup> + 1 *r ∂w ∂r* + *α*1 *w r ∂*2*w ∂r∂z* + *u r ∂*2*w ∂r*<sup>2</sup> + 3 *r ∂w ∂r ∂w ∂z* +1 *r ∂u ∂r ∂w ∂r* + 4 *∂w ∂r ∂*2*w ∂r∂z* + *w ∂*3*w ∂r*2*∂z* + 2 *∂u ∂r ∂*2*w ∂r*<sup>2</sup> +*u ∂*3*w ∂r*<sup>3</sup> + 3 *∂*2*w ∂r*<sup>2</sup> *∂w ∂z* + *∂*2*u ∂r*<sup>2</sup> *∂w ∂r* + *α*2 2 *r ∂u ∂r ∂w ∂r* + 2 *r ∂w ∂r ∂w ∂z* (2) +2 *∂*2*u ∂r*<sup>2</sup> *∂w ∂r* + 2 *∂u ∂r ∂*2*w ∂r*<sup>2</sup> + 2 *∂*2*w ∂r*<sup>2</sup> *∂w ∂z* +4 *∂w ∂r ∂*2*w ∂r∂z* + *β*3 2 *r ∂w ∂r* 3 + 6 *∂w ∂r* 2 *∂*2*w ∂r*<sup>2</sup> , 

$$
\mu \frac{\partial T}{\partial r} + \frac{w \partial T}{\partial z} = \frac{k}{\rho c\_p} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1 \partial T}{r \partial r} \right) + \frac{16 \sigma T\_\infty^3}{3k^\* \rho c\_p} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1 \partial T}{r \partial r} \right) + \frac{Q\_0}{\rho c\_p} \left( T - T\_\infty \right), \tag{3}
$$

$$\frac{\partial \mathcal{C}}{\partial r} + w \frac{\partial \mathcal{C}}{\partial z} = D \left( \frac{\partial^2 \mathcal{C}}{\partial r^2} + \frac{1 \partial \mathcal{C}}{r \partial r} \right),\tag{4}$$

with 
$$\begin{array}{ccccccccc}\hline \text{w (r,z)} & \mathcal{W}\_{\mathcal{W}}\left(z\right) = \frac{\mathcal{W}\_{\mathcal{W}}z}{\mathcal{I}}, \; \boldsymbol{u}\left(r,z\right) = \boldsymbol{0}, \; \boldsymbol{T}\left(r,z\right) = \boldsymbol{T}\_{0} + \boldsymbol{b}\left(\frac{z}{\mathcal{I}}\right), & \boldsymbol{C}\left(r,z\right) = \boldsymbol{C}\_{0} + \boldsymbol{d}\left(\frac{z}{\mathcal{I}}\right) \quad \text{at } r = \boldsymbol{R}, \\\ \boldsymbol{w}\left(r,z\right) & \longrightarrow & \mathcal{W}\_{\mathcal{C}}\left(z\right) = \frac{\mathcal{W}\_{\mathcal{C}}z}{\mathcal{I}}, \; \boldsymbol{T}\left(r,z\right) \to \boldsymbol{T}\_{0} + \boldsymbol{c}\left(\frac{z}{\mathcal{I}}\right), & \boldsymbol{C}\left(r,z\right) \to \boldsymbol{C}\_{0} + \boldsymbol{e}\left(\frac{z}{\mathcal{I}}\right) \quad \text{at } r \longrightarrow \infty. \end{array} \tag{5}$$

where *r* represents radial distance while *z* is assigned to axial distance. *u*, *v*, correspond to *r* and *z* component of fluid velocity, *T*, *T*∞ represent surface & ambient temperature while *C* and *C* ∞ represent

*u* surface & ambient concentration, respectively. Here *ρ*, *ν* correspond to fluid density and kinematic viscosity while *cp* and *k* are the specific heat at constant pressure and thermal conductivity, respectively. The constants *b*, *d*, *e* and *c* are dimensionless. *k*∗ is designated as coefficient of mean absorption, *σ* is Stephen Boltzman constant, reference length is represented by *l* , *Ww* is stretching velocity while *We* is the free stream velocity. *Q*0 is used to represent the coefficient of heat generation as well as absorption. It is pertinent to mention that *Q*<sup>+</sup> 0 (the positive values) behaves as source (heat generation) while *Q*− 0 (the negative values) behaves as sink (heat absorption). Using suitable transformations

$$\begin{array}{rcl} \text{aw}\left(r,z\right) &=& \frac{\text{W}\_{0}z}{\text{l}}f'\left(\eta\right), \text{ a}\left(r,z\right) = -\sqrt{\frac{\text{v}\overline{\text{W}\_{0}}}{\text{l}}}\frac{\text{R}}{\text{r}}f\left(\eta\right), \text{ } \eta = \sqrt{\frac{\text{W}\_{0}}{\text{v}\overline{\text{l}}}}\left(\frac{r^{2}-\overline{\text{R}}^{2}}{2\overline{\text{R}}}\right),\\ \theta(\eta) &=& \frac{T-T\_{\text{w}}}{\text{T}\_{W}-\text{I}\_{0}}, \ \phi(\eta) = \frac{\mathbb{C}-\mathbb{C}\_{\text{w}}}{\text{C}\_{w}-\text{I}\_{0}}. \end{array} \tag{6}$$

It can easily be verified that the balance of mass given by Equation (1) is identically satisfied. On substituting Equation (6) into Equations (2)–(5) and then rearranging we have:

> *f*

$$\begin{aligned} & \left(1 + 2\gamma\eta\right) f^{\prime\prime\prime} + A^2 + 2\gamma f^{\prime\prime} - f^2 + f f^{\prime\prime} \\ & + a\_1 \left[ \left(1 + 2\gamma\eta\right) \left\{ 2f^{\prime}f^{\prime\prime\prime} - f f^{\prime i\varepsilon} \right\} + 3f^{\prime\prime 2} \right] + \gamma \left(6f^{\prime}f^{\prime\prime} - 2f f^{\prime\prime\prime} \right) \\ & + a\_2 \left[ 2\left(1 + 2\gamma\eta\right) f^{\prime\prime 2} + \gamma \left(2f^{\prime}f^{\prime\prime} + 2f f^{\prime\prime\prime} \right) \right] \\ & + \beta \text{Re} \left[ \left(6\left(1 + 2\gamma\eta\right)^2 f^{\prime\prime} f^{\prime\prime} + 8\gamma\left(1 + 2\gamma\eta\right) f^{\prime\prime 3} \right] \right] \end{aligned} \tag{7}$$

$$f'(0) = 1, \; f\left(0\right) = 0, \; f'\left(\infty\right) = A,\tag{8}$$

$$\left(1+\frac{4}{3}\mathcal{R}\_d\right)\left(1+2\gamma\eta\right)\theta^{\prime\prime}+2\gamma\left(1+\frac{4}{3}\mathcal{R}\_d\right)\theta^{\prime}-\text{Pr}\left(f^{\prime}\theta+f^{\prime}\theta\right)-\text{S Pr}f^{\prime}+\text{Qg}=0,\tag{9}$$

$$
\theta(0) = 1 - S, \quad \theta(\infty) \longrightarrow 0,\tag{10}
$$

$$(1 + 2\gamma \eta) \phi'' + 2\gamma \phi' + \mathcal{S}c(f\phi' - f'\phi) - \mathcal{S}c\mathcal{S}t(f') = 0,\tag{11}$$

$$
\phi(0) = 1 - St\_{, \quad} \phi(\infty) \longrightarrow 0,\tag{12}
$$

where

$$\begin{array}{rcl} \mathfrak{a}\_{1} &=& \frac{a\_{1}^{\*}\mathcal{W}\_{0}}{l\mu}, \ \mathfrak{a}\_{2} = \frac{a\_{2}^{\*}\mathcal{W}\_{0}}{l\mu}, \ \ \beta = \frac{\beta\_{3}\mathcal{W}\_{0}^{2}}{l^{2}\mu}, \ \ \mathcal{R} = \frac{\mathcal{W}z}{\nu},\\ \mathcal{R}\_{d} &=& \frac{4\sigma^{\*}T\_{\infty}^{3}}{k^{\*}k}, \ \ \ \mathcal{P} = \frac{\mu c\_{p}}{\rho}, \ \gamma = \left(\frac{\nu l}{\mathcal{W}\_{0}R^{2}}\right)^{1/2},\\ \mathcal{Q} &=& \frac{\mathcal{Q}\overline{\rho}\_{0}}{\rho c\_{p}\mathcal{W}\_{w}}, \ \mathcal{S} = \frac{\epsilon}{b}, \ \ \mathcal{S}t = \frac{\epsilon}{d}, \end{array} \tag{13}$$

which respectively indicate the dimensionless third-grade parameters (*<sup>α</sup>*1, *α*2, *β*), the Reynolds number (*Re*), thermal radiation parameter (*Rd*), the Prandtl number (Pr), the curvature parameter (*γ*), heat generation/absorption parameter (*Q*), thermal stratification parameter (*St*) and solute stratification parameter (*Sc*). Expression of physical quantities are,

$$Nu\_{\overline{z}} = \frac{zq\_w}{k(T\_w - T\_0)}, \text{ where } \quad q\_w = -k \left(\frac{\partial T}{\partial r}\right)\_{r=R} \tag{14}$$

$$\mathcal{C}\_f = \frac{T\_{rz}}{\rho \mathcal{W}\_c^2} \tag{15}$$

$$Sh\_z = \frac{zh\_w}{D\_w(\mathbb{C}\_w - \mathbb{C}\_0)}, \text{ where } \quad h\_w = -D\left(\frac{\partial \mathbb{C}}{\partial r}\right)\_{r=R} \tag{16}$$

Such that

$$Nu\_{z}R\varepsilon\_{z}^{-1/2} = -(1+\frac{4}{3}Rd)(1-S)\theta'(0) \,,\tag{17}$$

$$\text{Re}\_z^{-1/2} \mathbb{C}\_f = [1 + 3\alpha\_1 + 3\beta \text{Re}(f''(0))^2] f''(0),\tag{18}$$

$$Sh\_z Re\_z^{-1/2} = -(1 - St)\phi'(0) \,, \tag{19}$$

where *Rez* = *Wwl*/*ν* is the local Reynolds number.
